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Chapter 3: Problem 30

Divide using synthetic division. $$\frac{x^{7}-128}{x-2}$$

Short Answer

Expert verified
The solution to the given division problem is \(x^6+2x^5+4x^4+8x^3+16x^2+32x+64 - \frac{64}{x-2}\)

Step by step solution

01

Setup the Synthetic Division

In order to use synthetic division, the divisor needs to be of the form \(x - c\) where \(c\) is a constant. That is the case here as \(x - 2\). The constant is 2. Set up the synthetic division by writing 2 to the left and the coefficients of \(x^{7}-128\) to the right. However, make sure to include zeroes for any missing powers of \(x\). So we rewrite the dividend as \(x^{7} + 0x^{6} + 0x^{5} + 0x^{4} + 0x^{3} + 0x^{2} + 0x -128 \)
02

Performing Synthetic Division

In synthetic division, first of all, bring down the first coefficient 1 (from \(x^7\)). Then multiply it with 2 (constant in \(x-2\)) and write the result under next coefficient of 0 from \(x^6\). Add these two numbers (0 obtained from multiplication, 0 from the polynomial) and write the result underneath. Repeat the process for all coefficients.
03

Write Out The Answer

The numbers on the bottom provide the coefficients for the powers of \(x\) of the quotient. The quotient is one degree less than the dividend (the polynomial we started with). The final answer will be \(x^6+2x^5+4x^4+8x^3+16x^2+32x+64 - 64/x-2\). Notice, that the remainder is negative, so it is subtracted

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Most popular questions from this chapter

Show that \(-1\) is a lower bound of \(f(x)=x^{3}-53 x^{2}+\) \(103 x-51 .\) Show that 60 is an upper bound. Use this information and a graphing utility to draw a relatively complete graph of \(f\).

Give an example of a function that is not subject to the Intermediate Value Theorem.

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

What do we mean when we describe the graph of a polynomial function as smooth and continuous?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-9 x^{2}$$
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